What are the most important things I need to do to encourage Google Sitelinks?

You will need to have quite a bit of traffic to your site to get those. Keep creating great content. Google will not give those out easy.

Be sure that you have your navigation marked up with semantic HTML and include an id of nav or navigation on your menu. This will help Google know what links are your navigation. (as opposed to a list of links).

Also, once you get them Google will give you an option to control what links you want in your sitelinks via Google Webmaster Tools.

There is no easy way or publicly know algorithm that will give you sitelinks. And I'm sure that algorithm changes all the time.

DO NOTS:

• Don't use JavaScript For links and menus.
• Don't think about it too much and give it time, sitelinks are rare.

Excerpt from Googler Maile Ohye's explanation on Sitelinks in Google Webmaster Help -

Sitelinks are often produced when there's a high probability that the site is the top match for the user's query. In other words, to have sitelinks ... you'll probably need to be the definitive first result

Sitelinks are completely automated & based on Google's algorithms. To get Sitelinks for a site it helps to have a well-structured website.

Sitelinks are usually taken from the home page. Make sure Google is able to see those links; if your website used heavy JavaScript to construct the DOM it's possible that Google simply doesn't see any link.

To see if this is the case open your website with a command line (no GUI) web browser, such as links (Linux). Alternatively, try to disable JavaScript in your own browser.